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Pi: History & Usages

Pi: History & Usages

A school presentation about Pi I made back in 2010.

Baptiste Fontaine

October 18, 2010
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  1. Introduction History Using π Calculate π Conclusion π History &

    Usages Baptiste Fontaine Paris Diderot University 18th of October 2010
  2. Introduction History Using π Calculate π Conclusion Introduction What’s Pi

    ? History Antiquity to Middle-Age Modern Age Using π Physics Math Calculate π With Geometry With Computer Conclusion
  3. Introduction History Using π Calculate π Conclusion What’s Pi ?

    Pi (π or Π) is an important mathematical constant, used in many science applications. The letter ”π” comes from the Greek alphabet. Three facts to know : • π is an irrationnal number, wich means its representation does not have an end. • π is the ratio of a circle’s circumference to its diameter. It is usually known as 3.141592... in decimal representation. • π is used since Antiquity
  4. Introduction History Using π Calculate π Conclusion What’s Pi ?

    Pi (π or Π) is an important mathematical constant, used in many science applications. The letter ”π” comes from the Greek alphabet. Three facts to know : • π is an irrationnal number, wich means its representation does not have an end. • π is the ratio of a circle’s circumference to its diameter. It is usually known as 3.141592... in decimal representation. • π is used since Antiquity
  5. Introduction History Using π Calculate π Conclusion What’s Pi ?

    Pi (π or Π) is an important mathematical constant, used in many science applications. The letter ”π” comes from the Greek alphabet. Three facts to know : • π is an irrationnal number, wich means its representation does not have an end. • π is the ratio of a circle’s circumference to its diameter. It is usually known as 3.141592... in decimal representation. • π is used since Antiquity
  6. Introduction History Using π Calculate π Conclusion Discovery of π

    and beginning of calculations (the following lists are not exhaustive) ∼ 2’600 BC Egyptian, Babylonian and Indian civilisations ∼ 900 BC use π for different constructions 434 BC Archimede tried to calculate approximately : π ≈ 3.1429 130 Zhang Heng finds π ≈ √ 10 : 3.162277... ... 800 Al Khwarizmi finds π ≈ 3.1416 1200 Fibonacci finds π ≈ 3.141818 ...
  7. Introduction History Using π Calculate π Conclusion Modern Age ...

    1424 Jamshid al-K¯ ashi finds 16 decimal places 1615 Ludolph van Ceulen finds 32 decimal places 1699 Abraham Sharp finds 71 decimal places 1841 William Rutherford finds 152 decimal places 1874 William Shanks finds 5270 decimal places ... 1982 Yoshiaki Tamura finds more than 2’000’000 decimal places ... 2010 Shigeru Kondo finds 5’000’000’000’000 decimal places
  8. Introduction History Using π Calculate π Conclusion Physics In Physics,

    π is used in some formulas, such as : • Coulomb’s law (electric force) F = |q1 q2 | 4π 0 r2 • Kepler’s third law constant (co-orbiting bodies) G(M + m) = ( 2π P )2a3 • Einstein’s Cosmological constant Λ = 8πGρvac
  9. Introduction History Using π Calculate π Conclusion Math Geometry In

    Geometry, π is used to calculate circumference (2πr, where r is the diameter) and area (πr2) of circles. Complexe numbers (in C) For example, a complex number z can be write like : az = r(cos φ + sin φ) And for i where i2 = −1 : eiπ = −1
  10. Introduction History Using π Calculate π Conclusion Calculate π using

    Geometry Archimede tried to calculate π with Geometry. Here is how we can do : 1. Let’s make an equilateral triangle. Every side mesure 1cm. 2. Then, we make a hexagone with six triangles. We can say that our hexagone is almost a circle. 3. We calculate the perimeter of the hexagon (p = 6 ∗ 1 = 6cm) and the longest diagonal (d = 2 ∗ 1 = 2cm). Then, we make a ratio of it : p d = 3
  11. Introduction History Using π Calculate π Conclusion Calculate π using

    Geometry Archimede tried to calculate π with Geometry. Here is how we can do : 1. Let’s make an equilateral triangle. Every side mesure 1cm. 2. Then, we make a hexagone with six triangles. We can say that our hexagone is almost a circle. 3. We calculate the perimeter of the hexagon (p = 6 ∗ 1 = 6cm) and the longest diagonal (d = 2 ∗ 1 = 2cm). Then, we make a ratio of it : p d = 3
  12. Introduction History Using π Calculate π Conclusion Calculate π using

    Geometry Archimede tried to calculate π with Geometry. Here is how we can do : 1. Let’s make an equilateral triangle. Every side mesure 1cm. 2. Then, we make a hexagone with six triangles. We can say that our hexagone is almost a circle. 3. We calculate the perimeter of the hexagon (p = 6 ∗ 1 = 6cm) and the longest diagonal (d = 2 ∗ 1 = 2cm). Then, we make a ratio of it : p d = 3
  13. Introduction History Using π Calculate π Conclusion • If we

    do it with a hectogon (100-sides regular polygon), the ratio is : p d ≈ 100 31.8362252 = 3.14107591
  14. Introduction History Using π Calculate π Conclusion Some Algorithms With

    computers, calculating π is faster. Here is some recursive algorithms. • Leibniz formula π 4 = ∞ n=0 (−1)n 2n + 1 = 1 − 1 3 + 1 5 − 1 7 ... • Ramanujan’s algorithm 1 π = 2 √ 2 9801 ∞ k=0 (4k)!(1103 + 26390k) (k!)43964k • Chudnovsky’s algorithm • Bailey’s algorithm • ...
  15. Introduction History Using π Calculate π Conclusion Some Algorithms With

    computers, calculating π is faster. Here is some recursive algorithms. • Leibniz formula π 4 = ∞ n=0 (−1)n 2n + 1 = 1 − 1 3 + 1 5 − 1 7 ... • Ramanujan’s algorithm 1 π = 2 √ 2 9801 ∞ k=0 (4k)!(1103 + 26390k) (k!)43964k • Chudnovsky’s algorithm • Bailey’s algorithm • ...
  16. Introduction History Using π Calculate π Conclusion Some Algorithms With

    computers, calculating π is faster. Here is some recursive algorithms. • Leibniz formula π 4 = ∞ n=0 (−1)n 2n + 1 = 1 − 1 3 + 1 5 − 1 7 ... • Ramanujan’s algorithm 1 π = 2 √ 2 9801 ∞ k=0 (4k)!(1103 + 26390k) (k!)43964k • Chudnovsky’s algorithm • Bailey’s algorithm • ...
  17. Introduction History Using π Calculate π Conclusion Some Algorithms With

    computers, calculating π is faster. Here is some recursive algorithms. • Leibniz formula π 4 = ∞ n=0 (−1)n 2n + 1 = 1 − 1 3 + 1 5 − 1 7 ... • Ramanujan’s algorithm 1 π = 2 √ 2 9801 ∞ k=0 (4k)!(1103 + 26390k) (k!)43964k • Chudnovsky’s algorithm • Bailey’s algorithm • ...
  18. Introduction History Using π Calculate π Conclusion Some Algorithms With

    computers, calculating π is faster. Here is some recursive algorithms. • Leibniz formula π 4 = ∞ n=0 (−1)n 2n + 1 = 1 − 1 3 + 1 5 − 1 7 ... • Ramanujan’s algorithm 1 π = 2 √ 2 9801 ∞ k=0 (4k)!(1103 + 26390k) (k!)43964k • Chudnovsky’s algorithm • Bailey’s algorithm • ...
  19. Introduction History Using π Calculate π Conclusion With Java We

    can try some algorithms with Java. For example, this is 1’000’000 iterations of Leibniz formula : double pi=0 ; for (int n=0 ;n¡=1000000 ;n++) pi+=(Math.pow(-1,n)/(2*n+1)) ; pi*=4 ; We find 3.14159. With 100’000’000 iterations, we find 3.14159265.